Exponential Thurston Maps and Limits of Quadratic Differentials

Contents 1. Introduction 2 Organization of the paper 3 2. The characterization theorem 4 2.1. Definitions and statement of the main theorem 4 2.2. Classification of postsingularly finite exponential maps 4 3. Iteration in Teichmüler space 6 3.1. The Teichmüler space of a topological exponential map 7 3.2. The Teichmüler metric and its dual 8 3.3. Examples of quadratic differentials 9 3.4. Proof of the Main Theorem 2.4 10 4. The decomposition theorem 13 4.1. The thick-thin decomposition theorem for quadratic differentials 14 4.2. Quadratic differentials on annuli 15 4.3. The component of a quadratic differential adapted to a short closed geodesic 17 4.4. The component of a quadratic differential adapted to the thick part 20 5. Limits of quadratic differentials 22 5.1. Limit models 22 5.2. Thin parts 25 5.3. Thick parts 27 6. Push-forward of quadratic differentials 30 Appendix A. Some general results on Riemann surfaces 35 A.1. Collars on hyperbolic Riemann surfaces 36 A.2. Mass per modulus is convex 37 Acknowledgements 39 References 39